Error

Target

Derivation

1. Split input into 3 regimes

2. if (tan eps) < -2.1880632665968565e-06

   1. Initial program 30.1

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum0.4

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied add-cube-cbrt0.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}\]

   6. Applied flip3--0.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

   7. Applied associate-/r/0.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

   8. Applied prod-diff0.8

      \[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) + \left(-\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_* + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*}\]

   9. Applied simplify0.4

      \[\leadsto \color{blue}{\left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)} + (\left(-\sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) + \left(\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right))_*\]

   10. Applied simplify0.4

       \[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]
   11. Using strategy rm

   12. Applied tan-quot0.5

       \[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - {\left(\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x\right)}^{3}} - \tan x\right) + 0\]

   13. Applied associate-*l/0.5

       \[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - {\color{blue}{\left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)}}^{3}} - \tan x\right) + 0\]

   14. Applied cube-div0.5

       \[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - \color{blue}{\frac{{\left(\sin \varepsilon \cdot \tan x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}}} - \tan x\right) + 0\]

   if -2.1880632665968565e-06 < (tan eps) < 3.601930261402066e-13

   1. Initial program 43.9

      \[\tan \left(x + \varepsilon\right) - \tan x\]

   2. Taylor expanded around 0 29.1

      \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]

   3. Applied simplify28.0

      \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*}\]

   if 3.601930261402066e-13 < (tan eps)

   1. Initial program 30.0

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum0.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied flip--0.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x\]

   6. Applied associate-/r/0.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]

   7. Applied fma-neg0.8

      \[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*}\]
3. Recombined 3 regimes into one program.

4. Applied simplify13.6

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\tan \varepsilon \le -2.1880632665968565 \cdot 10^{-06}:\\ \;\;\;\;\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left((\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right))_*\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \frac{{\left(\sin \varepsilon \cdot \tan x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{if}\;\tan \varepsilon \le 3.601930261402066 \cdot 10^{-13}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))